equation of conservative type arising in the

Hokkaido Mathematical Journal Vol. 32 (2003) p. 561-622

Singular limit of a second order nonlocal parabolic equation of conservative type arising in the micro-phase separation of diblock copolymers (Dedicated to Professors Masayasu Mimura and Takaaki Nishida on their sixtieth birthday)

M. HENRY, D. HILHORST and Y. NISHIURA (Received February 25, 2002; Revised August 15, 2002) Abstract. We study the limiting behavior as tends to zero of the solution of a second order nonlocal parabolic equation of conservative type which models the micr0-phase separation of diblock copolymers. We consider the case of spherical symmetry and prove that as the reaction coefficient tends to infinity the problem converges to a free boundary problem where the interface motion is partly induced by its mean curvature. \in

Key words: reaction-diffusion systems of conservative type, singular limits, nonlocal motion by mean curvature, asymptotic expansions.

1.

Introduction

In this paper, we consider a second order nonlocal parabolic equation of conservative type proposed by Ohnishi and Nishiura [9], namely

(\mathcal{P}^{\Xi})\{\begin{array}{l}u_{t}^{\Xi}--\triangle u^{\in}+\frac{1}{\epsilon^{2}}(f(u^{\mathcal{E}})-t_{\Omega}^{f(u^{6})}-\epsilon v^{6}) in\Omega\cross(0,T) (1.1)-\triangle v^{\in}--u^{\in}-t_{\Omega}^{u^{\in}} in\Omega\cross(0,T) (1.2)\frac{\partial u^{\mathcal{E}}}{\partial n}--\frac{\partial v^{\epsilon}}{\partial n}--0 in\partial\Omega\cross(0,T) (1.3)t_{\Omega}^{v^{\in}dx}--0 fort\in(0,T) (1.4)u^{\mathcal{E}}(x,0)--u_{0}^{\epsilon}(x) forx\in\Omega (1.5)\end{array}

where f(s):_{-}^{-}2s(1-s^{2}) ,

f_{\Omega}udx:-- \frac{1}{|\Omega|}\int_{\Omega}udx

1991 Mathematics Subject Classification : 35K57,35B25,53R35 .

562

M. Henry, D. Hilhorst and Y. Nishiura

and where is a smooth bounded domain. Integrating (1.1) in and using (1.4) we deduce that the integral of u is conserved in time, namely \Omega\subset R^{N}(N\underline{>}2)

\Omega

\int_{\Omega}u^{\epsilon}(x, t)dx--\int_{\Omega}u_{0}^{\epsilon}(x)dx--:\mathcal{M}_{0}^{\epsilon}

for all

(1.6)

t\in(0, T) .

Therefore is a second order model system which conserves mass. The main feature of this equation is that it shares the same stationary solutions as the fourth order model system arising in the micr0-phase separation of diblock copolymer melts (cf. [8]). Both problems are technically quite difficult: it is well-known that fourth order equations do not have a maximum principle which excludes making use of the usual techniques involving upper and lower solutions. The situation is similar for the conserved Allen-Cahn for which the maximum principle does not apply either and the only bounds which are known for the solutions can be obtained using arguments based on invariant domain. Moreover stability properties of those solutions also coincide with each other, namely Ohnishi and Nishiura [9] proved that the sign of the real parts of the spectrum corresponding to the fourth order model and to Problem coincide with each other. In general, it is much more difficult to study the spectrum distribution of the fourth order equation compared with the second order one, hence it is more informative to investigate than attacking the fourth order problem directly. As we shall discuss below, the singular limit equation of turns out to be a mean curvature flow with nonlocal term, which is much more intuitive than a free boundary problem of Mullin-Sekerka type associated to the fourth order problem. Note that the total mass does not change in polymer problems, hence the above conservation (1.6) is a natural consequence. We remark that the functional (\mathcal{P}^{\epsilon})

L^{\infty}

(\mathcal{P}^{\in})

(\mathcal{P}^{\epsilon})

(\mathcal{P}^{\epsilon})

(1.1)

E^{\epsilon}(u^{\Xi})-- \int_{\Omega}(\frac{\epsilon}{2}|\nabla u^{\epsilon}|^{2}+\frac{1}{\epsilon}F(u^{\epsilon})+\frac{1}{2}|\nabla v^{\epsilon}|^{2})dx

where F(s)– \frac{(1-s^{2})^{2}}{2} is a Lyapunov functional for Problem pose that the initial value satisfies the hypothesis

and sup-

(P^{\in})

u_{0}^{\in}\in H^{2}(\Omega)

that H_{0}^{\epsilon}\{\mathcal{M}_{0}^{\in}tendsto\mathcal{M}_{0}as\epsilon\downarrow 0thereexists\mathcal{M}_{0}\in(-|\Omega|,|.\Omega|)suchthatThereexistsapositiveconstantCsuch

H_{0}^{\epsilon}

,

E^{\epsilon}(u_{0}^{\in})\underline{0 , there exists b_{0}(\delta)>0 and K(\delta)>0 such that satisfying for and K(\delta) arrow 0 as the following properties and two (i) Suppose that there exist a time interval – – such that and t functions t a^{\in}and . , t) for all t\in[t_{1}, t_{2}] . , are two successives -zeros of a^{\in}on we have that the sign of Then denoting by b_{0}(\delta)

arrow\infty

\delta\downarrow 0

-

-

\epsilon\underline{}2(b_{0}(\delta)+1)

and

u^{\epsilon}(r_{+}^{\in}(t), t)

tanh

(6.2)

( \tau^{\epsilon}(\frac{r-r_{-}^{\in}(t)}{\epsilon})+\mu^{\epsilon})|\underline{